Problem Set N°2

Valentina Andrade

October 03, 2022

Abstract

The following report contains the exercises requested in problem set 2. In the first part you can download exercise solutions. In the second part, we study the Dornbush’s exchange rate overshooting hypothesis. Dornbusch’s overshooting hypothesis (1976) is an important tool for analyzing exchange rate and monetary policy dynamic. This report suggest a identification strategy (with VAR approach) that deals with simultaneity and long-run effects (neutrality assumption). Although previous research did not find clear effects (probably due to simultaneity biases), we found that a monetary policy shock has a strong and immediate effect on the exchange rate. Then, exchange rate gradually depreciates back to the baseline. In conclusion, the results are consistent with the overshooting hypothesis’

Part 2

Abstract

The overshooting hypothesis in the data

In 1976 Dornbusch present an important macroeconomic theory about expectations and exchange rate dynamic. Dornbusch’s overshooting exchange rate hypothesis sugest that an increase in the interest rate should cause the nominal exchange rate to appreciate instantaneously, and then depreciate in line with uncovered interest parity (UIP) (Bjørnland, 2009)

More precisly, in his model the dynamic aspects of exchange rate determination arise from the assumption that exchange rates and asset markets adjust rapidly relative to goods markets.

In particular, in the short run

  • A monetary expansion is shown to induce an immediate depreciation in the exchange rate and results therefore for fluctuations in the exchange rate and the terms of trade.

  • Rising prices may be accompanied by an appreciating exchange rate so that the trend behavior of exchange rates stands potentially in strong contrast with the cyclical behavior of exchange rates and prices.

  • There is a direct effect of the exchange rate on domestic inflation. The exchange rate is a critical channel for the transmission of monetary policy to aggregate demand for domestic output.

  • The effect of monetary policy on interest rates and exchange rates is significantly affected by the behavior of real output.

    • If real output is fixed, a monetary expansion will, in the short run, lower interest rates and cause the exchange rate to overshoot its long-run depreciation.
    • If output, responds to aggregrate demand, the exchange rate and interest rate changes will be buffered. While the exchange rate will still depreciate, it may no longer overshoot, and interest rates may actually rise.

However, only few empirical research have found support for Durnbusch hypothesis (see Sims (1992), Eichenbaum and Evans (1995) and Favero and Marcellino (2004) for Euro zona). An important problem that was adressed by these studies was the simultaneity between monetary policy and the exchange rate. To deal with it they used structural vector autoregressive models (VAR) (see Sims(1980)) by “placing recursive, zero contemporaneous restrictions on the interaction between monetary policy and exchange rate” (Bjørnland, 2009, p 65). In other word, we use VAR models by applying restrinctions that ensure a unique identification while contemporaneous interaction between monetary policy and the exchane rate.

Method

In order to conduct this, a Cholesky decomposition is applied to identify monetary policy shocks, that either

  1. Restricts monetary policy from reacting contemporaneously to an exchange rate shock

  2. Restricts exchange rate cannot react immediatly to a monetary policy shock (Favero and Marcellino, 2004)

The identification strategy consist in a \(Z_t\) vector of the variables discussed above \(Z_t = [i^*_t ~~~y_t ~~~ \pi_t ~~~ i_t ~~~ \triangle e_t]'\).

where - \(\pi_t\) = inflation

  • \(y_t\) = log of real gross domestic product

  • \(i^*_t\) = three-month domestic interest rate

  • \(i_t\) = trade-weighted foreign interest rate

  • \(\triangle e_t\) = log trade-weighted real exchange rate

The VAR is assumed to be stable (ie, no eigenvalues lies outside the unit circle) and can be inverted and represented in terms of its MA process

\[Z_t = C(L)\varepsilon_t\] where

  • \(\varepsilon_t\) are reduced form of structural shocks (\(\varepsilon\)~ iid\((o, \Omega)\)) (see Apendix 3)

  • \(C(L)= \sum_{j=0}^\infty C_j L^j\), \(B(L)S = C(L)\)

Finally, we can write the monetary policy shock \(\varepsilon^{MP}\) and exchange rate shock \(\varepsilon^{ER}\), assuming three zero restrictions on the relevant coefficents in the \(S\) matrix.

\[ \begin{bmatrix} i^*\\ y\\ \pi\\ i\\ \triangle e \end{bmatrix}=B(L) \begin{bmatrix} S_{11}&0&0&0&0\\ S_{21}&S_{22}& 0 & 0 &0\\ S_{31}&S_{32}&S_{33}&0&0 \\ S_{41}&S_{42}&S_{43}&S_{44}&S_{45}\\ S_{51}&S_{52}&S_{53}&S_{54}&S_{55} \end{bmatrix} \cdot \begin{bmatrix} \varepsilon^{i^*} \\ \varepsilon^Y \\ \varepsilon^{CP} \\ \varepsilon^{MP} \\ \varepsilon^{ER} \end{bmatrix} \]

Also, we can uniquely identified and orthogonalized the shocks if we say

\[B_{51}(1)S_{14} +B_{52}(1)S_{24}+ B_{53}(1)S_{34} +B_{54}(1)S_{44} + B_{55}(1)S_{54} = 0\]

Data

The model is estimated for Canada, which we have focused on because it is a small open country, as the exchange rate is an important transmission chanel for shocks. In particular, quarterly data from Q1 1994 to Q4 2021 are used (Central Bank of Canada). We do not use earlier starting period than 1994 because due to the comparability of data and it would make difficult to identidy a stable monetary policy regime (in 1983 experienced important structural changes) (see Clarida et al, 2000)

Figure 1. Trimestral series nominal_gdp, real_gdp, m1_money_supply, m2_money_supply, overnight_leading_rate, exchange_rate (1992- 2022)

As we can see in the Figure 1 some variables have trend (seassonal or linear). However, we carry out extensive robusteness test to the VAR specification (7 VAR models).

Also, the lag order of the VAR is determined using various information criteria, suggesting that threee lags are acceptable. In some specification are 4 and 3, but in order to compare the estimation we use 3 lags (instead, hypothesis of autocorrelation and heteroscedasticity is rejected) (Appendix 2)

Structural identification scheme

Figure 2. Canada: response to a monetary policy shock, using VAR approach and Bank of Canada data.

Figure 2 graph the impulse response of a monetary policy shock (in +100 basis point) (TPM) on interest rate, the level of the real exchange rate (TCR), GDP (PIB) and inflation (Inf). The upper and lower dashed lines plotted in each graph are probability bands. As we can see in the IRF \(TPM \Longrightarrow TPM\) a monetary policy shock increases interest rates temporarily. The interest rate returns to its steady-state after 1-2 years (4 to 8 quarters).

Futhermore, the monetary policy shock has a immediate and strong effect on the exchange rate, which is apreciated by 1% (the results have to be rescaled from percentage basis points to percent, which coincides with Bjørnland, 2009). However, there is some delay in overshooting for 4-5 months, before the exchange rate quickly depeciates to equilibrium. In other words, initial appreciation is small compared with the impact effect (gradually depreciation back to the baseline). This is consistent with other papers (see Zettelmeyer, (2004) with daily data; Kearns and Manners (2006) with intraday data; Cushman and Zha (1997) that fin instant overshooting in an analysis of Canada).

An importan issue is examine wheter there is any monetary response to exchange rate changes. If monetary policy reacts immediately to exchange rate variation, then we would expect that the interaction between interest rate and exchange rates is important to identify monetary policy shocks.

Figure 3. Canada: response to a exchange rate shock, using VAR approach and Bank of Canada data.

As we can see in Figure 3, an exchange rate shock that depreciates the exchange rate leads to produce a significant increase in the interest rate. Also, other studies show that the effect is largest in Canada than in other countries. For example, Bjørnland (2009) indicate that

“Canada displays by far the highest degree of interaction between interest rate and exchange rate dynamics, as monetary policy shocks explan 41% of the exchange rate variation on impact, while 52% of the interest rate variation is explained by exchange rate shocks on impact” (p. 70)

Robustnes of results

Now, we will discuss three dimensions with respect to the baseline specification:

1. Specification of the VAR

As suggest Table 1, we can use four instead of three lags in the VAR.

Table 1. VAR lag order selection criteria (see Canova, 2017)

lag logL LR p-val AIC SIC HQ
0 539.7989 NaN NaN -8.5774 -8.3954 -8.5035
1 1066.5545 1002.5348 0.0000 -16.8154 -16.2695 -16.5937
2 1157.6946 167.5803 0.0000 -18.0273 -17.1176 -17.6578
3 1171.9885 25.3601 0.0637 -17.9998 -16.7261 -17.4824
4 1186.1752 24.2548 0.0841 -17.9706 -16.3330 -17.3053
5 1206.6663 33.7110 0.0059 -18.0430 -16.0415 -17.2300
6 1218.4732 18.6625 0.2866 -17.9754 -15.6100 -17.0145
7 1222.5283 6.1481 0.9864 -17.7827 -15.0534 -16.6740
8 1234.9839 18.0807 0.3192 -17.7255 -14.6323 -16.4690

Figure 5 Canada: response to a monetary policy shock, using VAR approach and Bank of Canada data (dotted line corresponds to the estimation with 4 lags)

Figure 5 shows what happens to the IRF when the lags are changed. We can see that the difference in the response to the shock is not substantially different. Although at the beginning it is clear that the distance between the two estimates is different, however, as the quarters increase, this distance becomes shorter.

2. Choice of variables included in the VAR

Figure 6 Canada: response to a monetary policy shock, using VAR approach and Bank of Canada data

As a robustness exercise, 7 different models are estimated, each of them represented by the lines in Figure 6. The baseline model corresponds to the one between the red and purple zone. If we read the models from top to bottom

  • Model 2: The terms of trade are incorporated in the VAR.

  • Model 3: Internal and external terms of trade are incorporated.

  • Model 1: baseline

  • Model 4: in addition to what is indicated in model 3, the logarithm of the money supply M1 is added.

  • Model 5: As an alternative specification to inflation, Canada’s deflation is incorporated.

  • Model 7: in addition to what is indicated in model 5, the external rate (FED) is incorporated.

As in the exercise in Figure 5, we can see that with changes in the specification (incorporating or removing variables from the VAR) the response of TCR to a monetary policy shock does not change in any way. This same exercise was carried out with different specifications with and without trend variables and the results were similar.

3. Fitted values

Figure 7. Fitted value for baseline model

Figure 7 presents the evolution over time of the variables used in the baseline model (blue line) and their fitted values (red dotted line). As we can see, the model variables, i.e., government spending, inflation, overnight rate and real exchange rate, are predicted with a good fit. We can only identify that in the real exchange rate there is a small lag.

This result is consistent even when the specifications of the model are changed (see figures 3, 6,9,12,15,18,21 in the GitHub repository).

Conclusion

Dornbusch’s overshooting hypothesis (1976) is an important tool for analyzing exchange rate and monetary policy dynamic. This report suggest a identification strategy (with VAR approach) that deals with simultaneity and long-run effects (neutrality assumption). Although previous research did not find clear effects (probably due to simultaneity biases), we found that a monetary policy shock has a strong and immediate effect on the exchange rate. Then, exchange rate gradually depreciates back to the baseline. In conclusion, the results are consistent with the overshooting hypothesis.

References

Bjørnland, H. C. (2009). Monetary policy and exchange rate overshooting: Dornbusch was right after all. Journal of International Economics, 79(1), 64-77.

Canova, F., Paustian, M., 2007. Measurement with Some Theory: Using Sign Restrictions to Evaluate Business Cycle Models. Manuscript. Universitat Pompeu Fabra

Clarida, R., Galí, J., Gertler, M., 2000. Monetary policy rules and macroeconomic stability: evidence and some theory. Quarterly Journal of Economics 115, 147–180.

Cushman, D.O., Zha, T., 1997. Identifying monetary policy in a small open economy under flexible exchange rates. Journal of Monetary Economics 39, 433–448.

Dornbusch, R., 1976. Expectations and exchange rate dynamics. Journal of Political Economy 84, 1161–1176.

Eichenbaum, M., Evans, C., 1995. Some empirical evidence on the effects of shocks to monetary policy on exchange rates. Quarterly Journal of Economics 110, 975–1010

Favero, C.A., Marcellino, M., 2004. Large Datasets, Small Models and Monetary Policy in Europe. Manuscript, Bocconi University

Hamilton, J. D. (1994). Time series analysis. Princeton University Press. Chapters 1-3

Bockwell, P. J., & Davis, R. A. (1991). Time Series: Theory and Methods.. Chapter 3

Sims, C.A., 1980. Macroeconomics and reality. Econometrica 48, 1–48.

Sims, C.A., 1992. Interpreting the macroeconomic time series facts: the effects of monetary policy. European Economic Review 36, 975–1011.

Hayashi, F. (2011). Econometrics. Princeton University Press.

Zettelmeyer, J., 2004. The impact of monetary policy on the exchange rate: evidence from three small open economies. Journal of Monetary Economics 51, 635–652

Apendix A. The original model

A. Capital mobility and expectations

Short run

\[r = r^* + x\] where, \(r\) is the domestic interest rate, \(r^*\) interest rate of the world, \(x\) expected rate of depreciation of the domestic currency.

\(H_1\): If the domestic currency \(x\) is expected to depreciate, interest rate on assets will exceed those abroad by the expected rate of depreciation.

Long-run (perfect foresight expectations formation)

\[x = \theta (\bar e - e)\] States thar the expected rate of depreciation of the spot rate is proportional to the discrepancy between the long-run rate and the current spot rate.

B. The money market

Assuming a convetional demand for money, the log of which is linear in the log of real income and in interest rate we have

\[\lambda r + \phi y = m - p\] where \(m, p\) and \(y\) denote the logs of nominal quantity of money, price level, and real income. Combing the last three equations we can obtain the relationship between the spot exchange rate, price level and the long-run exchange rate given that the money market clears and net assets yields are equalized:

\[p - m = -\phi y + \lambda r^* + \lambda \theta (\bar e - e)\]

With some simplication we can obtain

\[e = \bar e - (1/ \lambda \theta)(p- \bar p)\] ### D. The Goods market

\[ln D = u + \delta (e-p) + y \gamma - \sigma r\] The rate of increase in the price of domestic goods \(\dot p\) can be written as

\[\dot p = \pi ln (D/Y) = \pi [ + \delta (e-p) + y \gamma - \sigma r]\]

Expectations

\[\bar \theta (\lambda, \delta , \sigma , \pi)=\pi(\sigma /\lambda + \delta)/2 + [\pi ^2 (\sigma/\lambda + \delta)^2/4 + \pi \delta /\lambda]^\frac{1}{2}\]

Appendix 3 - Residuals

Model 1

Model 2

Model 3

Model 4

Model 5

Model 6